Abstract
In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x β B β I from the interval I such as the inequality | P (x) | < Q-w, w > n, Q >1 for the polynomials P(x) β Z[x], deg P β€ n, H(P) β€Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w > n +1 we get the estimate Β΅ B< c1(n) Q β (w-1/n). The best estimate to date was c2(n) Q β(w- n/n).
Highlights
satisfied. The methods of obtaining estimates are different at different intervals of w change
Information about the authorΠΠ°ΡΠΈΠΌΠΎΠ²ΠΈΡ ΠΠ»Π΅Π½Π° ΠΠ°ΡΠΈΠ»ΡΠ΅Π²Π½Π° β Π°ΡΠΏΠΈΡΠ°Π½Ρ. ΠΠ½ΡΡΠΈΡΡΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΠΊΠΈ ΠΠΠ ΠΠ΅Π»Π°ΡΡΡΠΈ (ΡΠ». Π‘ΡΡΠ³Π°Π½ΠΎΠ²Π°, 11, 220072, ΠΠΈΠ½ΡΠΊ, Π Π΅ΡΠΏΡΠ±Π»ΠΈΠΊΠ° ΠΠ΅Π»Π°ΡΡΡΡ). E-mail: elena.guseva.96@ yandex.by. Bernik Vasiliy I. β D. Sc. (Physics and Mathematics), Professor, Chief researcher. Institute of Mathematics of the National Academy of Sciences of Belarus (11, Surganov Str., 220072, Minsk, Republic of Belarus). E-mail: bernik.vasili@ mail.ru
Summary
Π Π΅ΡΠ»ΠΈ a β ΠΈΡΡΠ°ΡΠΈΠΎΠ½Π°Π»ΡΠ½ΠΎΠ΅ ΡΠΈΡΠ»ΠΎ, ΡΠΎ Π½Π΅ΡΠ°Π²Π΅Π½ΡΡΠ²ΠΎ (2) ΠΈΠΌΠ΅Π΅Ρ Π±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ΅ ΡΠΈΡΠ»ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π² ΡΠ΅Π»ΡΡ ΡΠΈΡΠ»Π°Ρ p ΠΈ q. ΠΠΎΡΡΠΈ Π²ΡΠ΅ ΡΠΎΡΠΊΠΈ ΠΈΠ½ΡΠ΅ΡΠ²Π°Π»Π° β ΡΡΠΎ Π²ΡΠ΅ ΡΠΎΡΠΊΠΈ I Π±Π΅Π· ΡΠΎΡΠ΅ΠΊ x β B β I , ΞΌB =0, Π° 1(Ξ¨) β ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ x β I , Π΄Π»Ρ ΠΊΠΎΡΠΎΡΡΡ Π½Π΅ΡΠ°Π²Π΅Π½ΡΡΠ²ΠΎ xq - p < Ξ¨(q) ΠΈΠΌΠ΅Π΅Ρ Π±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ΅ ΡΠΈΡΠ»ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π² ΡΠ΅Π»ΡΡ p ΠΈ q. ΠΠ±ΠΎΠ·Π½Π°ΡΠΈΠΌ ΡΠ΅ΡΠ΅Π· n (Ξ¨) ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ x β I , Π΄Π»Ρ ΠΊΠΎΡΠΎΡΡΡ Π½Π΅ΡΠ°Π²Π΅Π½ΡΡΠ²ΠΎ. ΠΠ±ΠΎΠ·Π½Π°ΡΠΈΠΌ ΡΠ΅ΡΠ΅Π· n (Q, w) ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ ΡΠΎΡΠ΅ΠΊ x β I , Π΄Π»Ρ ΠΊΠΎΡΠΎΡΡΡ ΡΠ°Π·ΡΠ΅ΡΠΈΠΌΠΎ Π½Π΅ΡΠ°Π²Π΅Π½ΡΡΠ²ΠΎ. ΠΠ½ΡΠ΅ΡΠ²Π°Π»Ρ Οm (P) ΠΏΠΎΠ΄Π΅Π»ΠΈΠΌ Π½Π° ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΡΠ΅ ΠΈ Π½Π΅ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΡΠ΅: Π°) ΠΈΠ½ΡΠ΅ΡΠ²Π°Π» Ο m (P1), P1(x) β B1 = V (b ) n (Q) Π±ΡΠ΄Π΅ΠΌ Π½Π°Π·ΡΠ²Π°ΡΡ ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΡΠΌ, Π΅ΡΠ»ΠΈ Π΄Π»Ρ Π»ΡΠ±ΠΎΠ³ΠΎ Π΄ΡΡΠ³ΠΎΠ³ΠΎ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠ° P2 (x) β B1 Π²ΡΠΏΠΎΠ»Π½ΡΠ΅ΡΡΡ Π½Π΅ΡΠ°Π²Π΅Π½ΡΡΠ²ΠΎ ΞΌ(Ο m (P1). ΠΠ±ΠΎΠ·Π½Π°ΡΠΈΠΌ ΡΠ΅ΡΠ΅Π· B4 ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ x β Οm (R), Π΄Π»Ρ ΠΊΠΎΡΠΎΡΡΡ Π½Π΅ΡΠ°Π²Π΅Π½ΡΡΠ²ΠΎ (13) ΡΠ°Π·ΡΠ΅ΡΠΈΠΌΠΎ Π² ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠ°Ρ t1(x) ΡΡΠ΅ΠΏΠ΅Π½ΠΈ n1 ΠΈ Π²ΡΡΠΎΡΡ Q Ξ».
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